The modified Poincar´ e exponent
8.2. THE MODIFIED POINCAR ´ E EXPONENT OF A SEMIGROUP
always infinity even though there may be more geometric information to capture. In this section we introduce a modification of the Poincar´e exponent which agrees with the Poincar´e exponent in the case where G is strongly discrete, but can be finite even ifGis not strongly discrete.
We begin by defining the modified Poincar´e exponent of a locally compact groupG≤Isom(X). Letµbe a Haar measure on G, and for each s consider the Poincar´e integral
(8.2.1) Is(G) :=
Z
b−skgkdµ(g).
Definition8.2.1. Themodified Poincar´e exponent of a locally compact group
G≤Isom(X) is the number
e
δG =δe(G) := inf{s≥0 :Is(G)<∞}, whereIs(G) is defined by (8.2.1).
Example 8.2.2. LetX =Hd for some 2 ≤d <∞, and let G≤Isom(X) be a positive-dimensional Lie subgroup. Then Gis locally compact, but not strongly discrete. Although the Poincar´e series diverges for every s, the exponent of con- vergence of the Poincar´e integral (or “modified Poincar´e exponent”) is equal to the Hausdorff dimension of the limit set of G (Theorem 1.2.3 below), and so in particular the Poincar´e integral converges whenevers > d−1.
We now proceed to generalize Definition 8.2.1 to the case whereG≤Isom(X) is not necessarily locally compact. Fixρ >0, and consider a maximalρ-separated1 subsetSρ⊆G(o). Then we have
[
x∈Sρ
B(x, ρ/2)⊆G(o)⊆ [ x∈Sρ
B(x, ρ),
and the former union is disjoint. Now suppose that G is in fact locally compact, and letν denote the image of Haar measure onGunder the mapg7→g(o). Then iff is a positive function onX whose logarithm is uniformly continuous, we have
X x∈Sρ f(x)≍×,ρ,f X x∈Sρ Z B(x,ρ/2) f dν ≤ Z f dν ≤ X x∈Sρ Z B(x,ρ) f dν≍×,ρ,f X x∈Sρ f(x).
Thus in some sense, the counting measure on Sρ is a good approximation to the measureν. In particular, takingf(x) =b−kxk gives
Is(G)≍×,ρ
X
x∈Sρ b−kxk.
1Here, as usual, aρ-separated subset of a metric spaceX is a setS⊆X such that d(x, y)≥ρ
for any distinctx, y∈S. The existence of a maximalρ-separated subset of any metric space is guaranteed by Zorn’s lemma.
Thus the integralIs(G) converges if and only if the seriesPx∈Sρb−kxk converges. But the latter series is well-defined even ifGis not locally compact. This discussion shows that the definition of the “modified Poincar´e exponent” given in Definition 8.2.1 agrees with the following definition:
Definition8.2.3. FixGIsom(X).
• For each setS⊆X ands≥0, let Σs(S) = X x∈S b−skxk ∆(S) ={s≥0 : Σs(S) =∞} δ(S) = sup ∆(S). • Let (8.2.2) ∆eG = \ ρ>0 \ Sρ ∆(Sρ),
where the second intersection is taken over all maximalρ-separated sets
Sρ.
• The number eδG = sup∆eG is called the modified Poincar´e exponent of
G. IfeδG ∈∆eG, we say thatGis ofgeneralized divergence type,2while if
e
δG∈[0,∞)\∆G, we say thatGis of generalized convergence type. Note that if eδG =∞, thenGis neither of generalized convergence type nor of generalized divergence type.
The basic properties of the modified Poincar´e exponent are summarized as follows:
Proposition8.2.4. FixGIsom(X). (i) ∆eG ⊆∆G. (In particularδeG≤δG.) (ii) If Gsatisfies
(8.2.3) sup
x∈X
#{g∈G:d(g(o), x)≤ρ}<∞ ∀ρ >0,
then∆eG= ∆G. (In particularδeG=δG.)
(iii) IfeδG<∞, then there existρ >0and a maximalρ-separated setSρ ⊆G(o) such that #(Sρ∩B)<∞ for every bounded setB.
(iv) For all ρ > 0 sufficiently large and for every maximal ρ-separated set
Sρ⊆G(o), we have ∆(Sρ) =∆eG. (In particularδ(Sρ) =eδ(G).)
2We use the adjective “generalized” rather than “modified” because all groups of conver-
gence/divergence type are also of generalized convergence/divergence type; see Corollary 8.2.8 below.
8.2. THE MODIFIED POINCAR ´E EXPONENT OF A SEMIGROUP 133
Remark 8.2.5. If G is a group, then it is clear that (8.2.3) is equivalent to the assertion that Gis strongly discrete. IfGis not a group, then by analogy we will say thatGis strongly discrete if (8.2.3) holds. (Recall that in Chapter 5, the various notions of discreteness are defined only for groups.)
Proof of Proposition 8.2.4.
(i) Indeed, for everys≥0,ρ >0, and maximalρ-separated setSρ we have Σs(Sρ)≤Σs(G) and thus∆(e G)⊆∆(Sρ)⊆∆(G).
(ii) Fix ρ > 0, and let Sρ ⊆G(o) be a maximal ρ-separated set. For every
x∈G(o) there existsyx∈ Sρ with d(x, yx)≤ρ. Then for eachy ∈ Sρ, we have
#{x∈G(o) :yx=y} ≤Mρ,
whereMρ is the value of the supremum (8.2.3). Therefore for eachs≥0 we have Σs(G) = X x∈G(o) b−skxk≍× X x∈G(o) b−skyxk ≤Mρ X y∈Sρ b−skyk=MρΣs(Sρ). In particular, Σs(G)<∞if and only if Σs(Sρ)<∞, i.e. ∆(G) = ∆(Sρ). Intersecting overρ >0 andSρ⊆G(o) yields ∆(G) =∆(e G).
(iii) TakeρandSρ such thatδ(Sρ)<∞. Before proving (iv), we need a lemma:
Lemma 8.2.6. Fix ρ1, ρ2 >0 with ρ2≥2ρ1. Let S1⊆G(o) be a ρ1-net,3and
letS2⊆G(o) be aρ2-separated set. Then
(8.2.4) ∆(S2)⊆∆(S1).
Proof. Since S1 is a ρ1-net, for every y ∈ S2, there exists xy ∈ S1 with d(y, xy)< ρ1. Ifxy =xz for somey, z∈S2, thend(y, z)<2ρ1≤ρ2 and sinceS2
is ρ2-separated we havey =z. Thus the mapy 7→xy is injective. It follows that for everys≥0, we have
Σs(S2) = X y∈S2 b−skyk≍× X y∈S2 b−skxyk≤ X x∈S1 b−skxk= Σs(S1), demonstrating (8.2.4). ⊳
(iv) The statement is trivial ifeδG=∞. So suppose thatδeG<∞, and letρ,Sρ be as in (iii). Fixρ′≥2ρand a maximalρ′-separated setS
ρ′ ⊆G(o), and
we will show that ∆(Sρ′) = ∆eG. The inclusion ⊇ follows by definition.
3Here, as usual, aρ-net in a metric spaceXis a subsetS⊆X such thatX=Nρ(S). Note that
To prove the reverse direction, fixρ′′>0 and a maximalρ′′-separated set
Sρ′′, and we will show that ∆(Sρ′)⊆∆(Sρ′′).
LetF=Sρ∩B(o, ρ′′+ρ); then #(F)<∞. We then set
Sρ′ :=
[
x∈Sρ′′
gx(F)
where for eachx∈Sρ′′,x=gx(o). Then for alls≥0, Σs(Sρ′′) = X x∈Sρ′′ b−skxk≍× X x∈Sρ′′ X y∈F b−skxk ≍× X x∈Sρ′′ X y∈F b−skgx(y)k = Σs(Sρ′)
and therefore ∆(Sρ′′) = ∆(Sρ′). ButSρ′ is aρ-net, so by Lemma 8.2.6, we
have ∆(Sρ′)⊆∆(Sρ′). This completes the proof.
Combining with Observation 8.1.5 yields the following:
Corollary 8.2.7. Suppose thatGis a group. If ∆6=∆e then
e
δ < δ=∞.
Corollary 8.2.8. If a group G is of convergence or divergence type, then it is also of generalized convergence or divergence type, respectively.
We will call a groupG≤Isom(X)Poincar´e regular if∆eG= ∆G, andPoincar´e irregular otherwise. A list of sufficient conditions for Poincar´e regularity is given in Proposition 9.3.1 below. Conversely, several examples of Poincar´e irregular groups may be found in Section 13.4.
CHAPTER 9